Smoluchowski Model Research Articles

Kinetics of magnetic flocculation. II. Flocculation of coarse particles

For pt.I see ibid., vol.23, p.1447 (1990). In theoretical studies of flocculation kinetics one usually assumes that the flocculation process reaches a steady state very rapidly. This assumption is invalid when a long-range attraction between the particles is present. In this paper the authors show that such a situation occurs for magnetic flocculation, provided that the particles are not too small. A full mathematical description of such a process is very difficult. As an alternative, a modified version of von Smoluchowski's model for rapid flocculation is presented, which describes the qualitative features of their experimental observations of aqueous Mn2O3 colloids with a particle diameter of about 0.8 mu m quite well. In particular this model suggests that the deviations from the 'steady-state' predictions increase very rapidly with increasing particle size, making these results almost always incorrect for many practical applications of magnetic flocculation.

Fractal structure of hematite aggregates

The structure of hematite aggregates, considered to behave as fractal objects, is studied using static light scattering methods. The fractal dimension obtained from the scattering exponent is found to be dependent on the aggregation mechanism and it ranges from 2.3 for rapid (diffusion limited) to 2.8 for slow (reaction limited) aggregation. Polydispersity and restructuring of aggregates do not affect the relationship between scattering exponent and fractal dimension in these aggregates. Excellent correspondence over a range of temperatures and ionic strengths is obtained between results of experiments using dynamic light scattering to determine aggregation kinetics and those predicted using a modified Smoluchowski model incorporating fractal dimensions.

Cluster and barrier effects in the temperature and pressure dependence of the photoisomerization of t r a n s-stilbene

The pressure and temperature dependence of the photoisomerization rate coefficient of trans-stilbene in the S1 state have been measured in the solvents C2H6, C3H8, C4H10, Xe, Co2, SF6, and CHF3. At constant temperature, the pressure dependences up to 6 kbar can be well represented by the Kramers–Smoluchowski model. The comparison of results in different solvents clearly indicates the importance of reactant–solvent cluster formation modifying the height and imaginary frequency of the barrier. The change of the temperature dependence with pressure points towards a multidimensional barrier of nonseparable character. Multidimensional barrier effects manifest themselves most clearly via the temperature dependence of the rate coefficient in the Kramers–Smoluchowski limit.

A generalized radiation boundary condition

The radiation boundary condition was originally proposed by Collins and Kimball as a means of avoiding the unphysical prediction of the Smoluchowski model for reaction rates that the calculated rate in three dimensions k(t) has the property k(0)=∞. A microscopic model that can be used to derive the boundary conditions uses the tacit assumption that an encounter between two molecules A and B gives rise to a reaction with a probability α<1. We consider a non-Markovian model in which the probability that exactly n encounters between A and B are required to produce a reaction is equal to θn. We show that when the expected number of such encounters is finite, one gets the usual radiation boundary condition. When the expected number is infinite, one finds a boundary condition that is nonlocal in time. The extension of our analysis to higher dimensions as well as to the Smoluchowski equation is readily generated.

Fractal Concepts And Aggregation Of Iron Oxides

ABSTRACTThe modelling of the aggregation kinetics of iron oxides has been succesful in predicting the increase in aggregate size as determined by dynamic light scattering measurements. The aggregates were found to exhibit fractal behaviour with fractal dimensions obtained from the scattering exponent in static light scattering studies dependent on the aggregation mechanism and ranging from 2.3 for rapid (diffusion limited) to 2.8 for slow (reaction limited) aggregation. Polydispersity and restructuring of aggregates were found not to affect the relationship between scattering exponent and aggregate fractal dimension. Excellent correspondence over a range of temperatures and ionic strengths has been obtained between results of sizing experiments using dynamic light scattering and sizes predicted using a modified Smoluchowski model incorporating fractal dimensions.

Aggregation kinetics in model colloidal systems: A light scattering study

We report an extensive set of quasi-elastic light scattering measurements of the hydrodynamic radius R H of aggregating polystyrene latices. The experiments have been performed in the transient regime in order to follow the kinetics of the aggregation process induced by the addition of a simple salt. We are able to distinguish various regimes of aggregation in different physical situations. We observe reversible flocculation and diffusion limited cluster aggregation (DLCA) in latices of 481 nm diameter, but also reaction limited cluster aggregation (RLCA) followed by DLCA for particles of 91 nm. The growth laws of R H in time have been verified and give exponential RLCA kinetics for short times and power law DLCA kinetics for long times. Both the exponential and the power law behaviors are observed in the intermediate time regime. In the DLCA case, for the largest particles we used, deviations from the asymptotic law have been obtained and can be described in terms of the Smoluchowski model for Brownian aggregation, together with the fractal nature of the resulting aggregates.

Photoemission from a spherical particle suspended in an electric field. The Smoluchowski model

In the study of photoemission from particles suspended in an ambient gas by an electric field, the fraction of electrons captured by an external electrode is determined as a function of the magnitude of the balance field (which is inversely proportional to the charge on the emitting particle). The experimental results have been interpreted previously in terms of a model which examines the ‘window’ for escape of the electron through the electrostatic field distribution. Here the diffusion of the electron within the gas is also taken into account by noting that the principal features of the problem are similar to those of the well-known Onsager problem of the relative escape of a geminate pair of ions in an electric field.

Diffusion coefficients of quenchers in proteins from transient effects in the intensity decays.

We used 2-GHz frequency-domain fluorometry to examine the intensity decays of N-acetyl-L-tryptophamide (NATA) and the protein staphylococcal nuclease in the presence and absence of quenching by oxygen or acrylamide. When analyzed with a multiexponential model, the decays of NATA and nuclease both become more heterogeneous in the presence of quenching. We attribute the increased complexity to transient effects in quenching or equivalently a time-dependent rate constant for quenching. The frequency-domain data were analyzed using the Smoluchowski model (exp(-t/tau-2b square root t)) and the radiation model, which is known to correct some flaws in the more approximate Smoluchowski model. The radiation model provides improved fits to the data, as evidenced by average 10-fold decreases in chi R2. The radiation model also provides an estimate of the sum of the diffusion coefficients and the specific rate constant for quenching. The apparent diffusion coefficients for acrylamide and oxygen in nuclease, as seen by its single tryptophan (residue 140) are 15- and 11-fold lower than in water, respectively. The apparent values of the oxygen diffusion coefficient in water, as seen by NATA, are 2- to 3-fold larger than expected from earlier steady-state measurements. The ability to recover the detailed form of the intensity decays by the frequency-domain method should allow comparison of experimental results with calculated trajectories of quenchers in proteins.

Radiation effects on phase separation and viscosity in a B2O3-PbO glass

Taking into account the temperature rise under the electron beam, phase separation in a supersaturated B2O3-PbO glass is shown to be enhanced by electron irradiation. Coagulation of PbO-rich particles is observed during in situ electron microscopic experiments. This mechanism of coalescence is discussed. Using the Smoluchowski model of coagulation we show that the glass viscosity is drastically lowered: typically eight to nine orders of magnitude at 270 °C and only one order at 420 °C. This effect is shown to disappear almost instantaneously when irradiation is stopped.

Coagulation in cell suspensions: Extensions of the von Smoluchowski model

Recently, a long-range interactive force between erythrocytes has been proposed (Rowlands et al., 1981, 1982a,b) based on an apparent increase in the rate of aggregation of erythrocytes in an aqueous suspension over that predicted by one model of Brownian aggregation. Here, we examine the assumptions underlying this model and propose modifications compatible with the biological constraints on the model. The refined model is represented as a series of coupled differential equations representing the change in particle number density as a function of time. Numerical solution of these equations is consistent with the absence of an intercellular force.

Overview of theoretical models for reaction rates

There is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences. In this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the National Institutes of Health on May 6–8, 1985. This review will focus on (1) the Smoluchowski model for reaction rates together with its extension by Onsager, (2) first passage time formalism for discrete and continuous master equations and Fokker-Planck equations, (3) the Kramers model and its extensions, (4) diffusion in the presence of trapping centers.

Viscosity dependence of the kinetics of the diffusion-controlled reaction of carbon monoxide with the separated alpha and beta chains of hemoglobin.

The kinetics of the recombination reaction of carbon monoxide with the isolated alpha and beta chains of human hemoglobin have been examined by laser flash photolysis in glycerol-water as a function of temperature and solvent viscosity. The second-order recombination rate constant is inversely proportional to viscosity raised to the 0.5 power--paralleling that predicted for the CO diffusion coefficient. This viscosity exponent is independent of the protein. These results are consistent with the reaction kinetics being essentially diffusion controlled in the high viscosity glycerol-water. For the alpha and beta chains, respectively, the diffusion-controlled rate constant is 0.003 and 0.002 of that predicted from the simple von Smoluchowski model based on the diffusion coefficients and molecular sizes of uniformly reactive spherical molecules. Several models incorporating steric requirements are used to rationalize the results. These models indicate that steric requirements for reaction in the diffusion-controlled limits are not greatly different in the alpha and beta chains and are only slightly less stringent than for myoglobin.

Kinetics of formation of the electron donor–acceptor complex between iodine and imidazole in chlorobutane solution

The kinetics of the formation of the electron donor–acceptor complex between iodine and imidazole in 1-chlorobutane at –64 °C have been investigated by an improved microwave temperature-jump apparatus. The forward rate constant kf is (3.3 ± 0.4)× 109 dm3 mol–1 s–1. This is nearly the same as the diffusion-controlled value kD calculated from the simple Smoluchowski model for reaction at every encounter between spherical molecules, without geometrical requirements, in a homogeneous fluid (kf/kD= 0.8).

Flocculation of brackish water from a tidal river

The control of the flocculation of water from tidal rivers is difficult because the water characteristics oscillate rapidly. Especially ionic strength and suspended solids concentration may have 10-fold changes in less than 6 h. Hence a relationship between an optimal dose of coagulant and raw water characteristics was sought with the aid of jar-tests. A pH of 6 was shown to be an acceptable value for flocculation with iron chloride and alum. The influence of change of ionic strength was found to be negligible. Suspended solids concentration proved to be a decisive control parameter. The von Smoluchowski model is appropriate to describe the kinetics of this sweep coagulation.

Smoluchowski's theory of coagulation in colloids holds rigorously in the Boltzmann-Grad-limit

We consider Smoluchowski's model of coagulation in colloids: n particles move in three-dimensional euclidean space according to Brownian motions independently of each other as long as the particles are at a distance greater than R. When two particles come to within a distance R they stick together and form a “double particle”, which itself is in Brownian motion — and so on. In the Boltzmann-Grad-limit n→∞, n R=constant, we prove “propagation of chaos” and derive the kinetic equations for the densities of the k-fold particles.

Floc simulation: Prolate spheroidal particles

In order to extend the usefulness of the simulation models a study is made of the structure of the flocs formed from prolate spheroidal primary particles of different elongations. Each spheroidal particle is represented by a group of tangent spheres inside the spheroidal envelope. By treating each group of spheres as a preformed cluster and using the improved Smoluchowski model a large number of flocs of different sizes and from different particle shapes were generated. The structure of these flocs was thoroughly investigated and the morphological parameters representing the form of an average floc of each species were calculated. The results were accumulated in the form of graphs and, in some cases, one representative graph from each series was selected to be used as an example in this manuscript.

Kinetics of Aggregation of Human Platelets

A study of the aggregation kinetics of human platelets using an electronic counting device is reported. The experimental data were analyzed quantitatively by a physical model, which assumed that the initial disappearance rate of single platelets versus time fitted a second-order type of aggregation with respect to platelet number. The mechanism of aggregation was surface barrier controlled. Thus, the aggregation rate constants in different adenosine diphosphate concentrations (1.5–9.0 μg) were 10–100 times greater than the rate constant (6.6325 × 10–12cm3/sec) for a diffusion-controlled process as calculated from Smoluchowski's model. Also, the stability constant values, which were equal to unity in a fast diffusion-controlled mechanism, were smaller in the surface-barrier-controlled process and ranged from 0.00741 to 0.0467. The extent of aggregation was indicated by the calculation of a sticking probability constant as determined by the barrier. Adenosine diphosphate induced a rapid aggregating effect. Prostaglandin El produced the most drastic deaggregating effect as compared to dinoprostone (prostaglandin E2) and dinoprost (prostaglandin F2α). Aspirin completely blocked the aggregating effect of arachidonic acid.

Kinetics of partly diffusion controlled reactions. I. Transient and apparent transient effect in fluorescence quenching

Analysis of decay curves of electronically excited molecules A* versus time in presence of quencher B leads to the determination of kinetic data for the reaction (i.e. for diffusion limited reactions, the experimental collisional distance σ′ greater than the true collisional distance σ) and the sum D of diffusion coefficients of both reactants). Experimental fluorescence steady state measurements (Stern-Volmer representation) are inconsistent with calculated curves using classical Smoluchowski's model with σ′and D. The difference between this has been interpreted by complex formation in the ground state between A and B. But, this assumption is unnecessary if we take into account a “static” quenching arising from B molecules located at distances between σ and σ′ from A*. A theoretical model based upon this principle is described; good agreement between the model and experimental results was obtained.

Fourier inversion of light scattering intensity data from coagulating dispersions

The short range structure of aggregates, which are formed during the coagulation of monodisperse sols, is investigated by developing a rigorous approach to the analysis of light scattering data. The particle distribution function is determined by the Fourier transformation of scattered light intensity data measured over scattering angles in the range 28 to 150°. The direct inversion of the intensity data produces large oscillations in the particle distribution function owing to the truncation of the data set as characterised by the “cut-off” parameter β(∝ particle size and incident light wavelength). These oscillations are removed by including an exponential modification function in the kernel of the transform. The effect of both the “cut-off” function and the “damping” function are assessed by adopting a general analysis and then testing the transformation method on intensity data generated using the Smoluchowski–Benoit model to simulate light scattering from a coagulating system. The results of an experimental investigation of the coagulation of a 0.234 µm diameter polystyrene sol indicates the formation of close-packed aggregates with the average number of neighbours increasing only slightly faster than expected from the von Smoluchowski model of coagulation. The average number of nearest neighbours is considerably less than expected from the Smoluchowski–Benoit model assuming the particles to be hexagonally close-packed.

Two dimensional diffusion theory: Cylindrical diffusion model applied to fluorescence quenching

A value for the diffusion constant of fluorescent quenchers can be inferred by fitting measured and theoretical fluorescence decay curves. In this paper, a formalism is presented describing diffusion-dependent fluorescence quenching in systems which are effectively two dimensional in nature. This model predicts somewhat different values for the diffusion constant from those one obtains by fitting the same experimental curve to the usual Smoluchowski model for diffusion in an isotropic (’’spherical’’) system.